Abstract
We study the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazanov-Stroganov Lax operator. The results apply as well to the spectral analysis of the lattice sine-Gordon model with integrable open boundary conditions. This spectral analysis is developed by implementing the method of separation of variables (SoV). The transfer matrix spectrum (both eigenvalues and eigenstates) is completely characterized in terms of the set of solutions to a discrete system of polynomial equations in a given class of functions. Moreover, we prove an equivalent characterization as the set of solutions to a Baxterâs like T-Q functional equation and rewrite the transfer matrix eigenstates in an algebraic Bethe ansatz form. In order to explain our method in a simple case, the present paper is restricted to representations containing one constraint on the boundary parameters and on the parameters of the Bazanov-Stroganov Lax operator. In a next article, some more technical tools (like Baxterâs gauge transformations) will be introduced to extend our approach to general integrable boundary conditions.
Highlights
IntroductionThe study of quantum models with integrable open boundary conditions has attracted a large research interest, e.g. see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52] and references therein
We define the most general cyclic representations of the 6-vertex reflection algebra associated to the Bazhanov-Stroganov Lax-operator
Qâ is the coefficient in Î(pâ1)N of the power expansion of the polynomial Q(λ). Once this notation are introduced, the previous characterization of the spectrum can be reformulated in terms of Baxterâs type TQ-functional equations and the eigenstates admit an algebraic Bethe ansatz like reformulation, as we show in the theorem
Summary
The study of quantum models with integrable open boundary conditions has attracted a large research interest, e.g. see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52] and references therein. While its dynamics has been studied by the exact computation of correlation functions first in [14, 15] and in [16], there generalizing in the ABA framework the method established in [114, 115] for the periodic chains These open quantum spin chains with z-oriented boundary magnetic fields correspond in the Sklyaninâs construction to the diagonal scalar solution of the reflection equation. A different approach, based on the generalization of the Sklyaninâs separation of variables (SoV) method to the reflection algebra framework, has lead to the complete eigenvalues and eigenstates characterization [29, 30, 33,34,35,36], proving its equivalence to an inhomogeneous TQ functional equation [36], and giving access to first computations of matrix elements of local operators [33] in the eigenstates basis. The remaining three appendices deal with the reduction of our representations to those associated to the chiral-Potts, the sine-Gordon and the XXZ spin s-chains at the 2s+1 roots of unit
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